Good question! Although the Higgs field gives rise to the masses of fundamental particles, it is not responsible for the gravitational force between massive particles. In other words, due to spontaneous symmetry breaking, the Higgs field causes mass terms in the Lagrangians for the effective field theories of the fermions in the standard model. However, the resulting effective field theories do not describe the gravitational force. Somehow, gravity then uses these masses to produce an attraction (actually it uses these masses to curve spacetime according to general relativity). This gravitational mechanism is not currently incorporated into a unified theory that involves all the forces of nature.
Based on the comments, it seems that I may have misunderstood the question of the OP. So here, I'll try to give a better picture:
Above the high energy level where the electroweak symmetry breaking occurs (electroweak scale), the Higgs field (more than just the Higgs boson) exists as a scalar field that couples to the fermion fields via Yukawa couplings.
Below the electroweak scale, the Higgs field gives rise to a vacuum expectation value (VEV), which turns the Yukawa coupling terms into mass terms in the Lagrangian. The only remaining Yukawa coupling terms is the one with the Higgs boson, which itself also acquires a mass.
Now to clear up some misconception. If we ignore gravity (as the OP seems to want to do) then there is no force between masses. In other words, the Yukawa coupling term that survives beneath the electroweak scale cannot be interpreted as an interaction for which the mass represents the coupling strength. The reason is simple, consider for instance the electromagnetic interaction: its interaction term in the Lagragian contains a coupling constant namely the electric charge that tells us how strong the force is. By contrast, the coupling constant associated with the Yukawa coupling with the Higgs boson is not the mass of the. In fact the mass terms and the Yukawa coupling terms are different terms in the Lagragian. It is true that the Yukawa couplings help to determine the values of the different masses, but that does not mean that the masses play a role in the interation.
It should also be mentioned that the remaining Yukawa coupling gives rise to an extremely weak force, due to the large mass of the Higgs boson; weaker even than the weak nuclear force, which gives us beta decay. It is effectively a contact force that one can only really observe a very high energies.
In summary, one should not confuse the Yukawa coupling associated with the Higgs boson as a force between masses (such as gravity).